How to make my logo spin showing the front side on both sides?
it seems you want to show the same image on front and back (we don't know if as in a coin, ie the same exact image on both sides, or as a photographic film, ie the backside looks reversed ). 

This is also a multiple question, as different steps are needed. So I will just drive you through what you need to learn to do.

Since you import a svg you get a curve object: it must be converted to mesh object first (alt+c), then add a face. If it needs depth, you can extrude that face (and so you get two faces, one on each side).

then you must assign the image to the face(s). This can be done in a number of ways (uvmap, different materials...) and it also can depend on the rendering engine used... 

Lastly you need the rotation. This is done adding a few (rotation) keyframes at different frames, enough for one full turn. by default you get an "eased" movement (starts and stops slowly) but you can later edit animation curves to make it constantly rotating, without "easing", if you wish so...

As you see, you have different (simple) tasks to learn. As there are already many answers for these tasks, look for them with the search tool...

edit: I noticed you added an example image but it seems not symmetrical, so it won't be possible to show the exact image on both sides, apparently. To explain better, I created an example with a (even more) asymmetrical mummy image and imported svg curve, like this:



here 


the svg has been converted (ALTC) to mesh, and rotated 90°over X
then selecting all vertices I created a face (F). 
then I extruded it over Y (EY) a bit to give it depth. 
then in edit mode, ortho mode, I created a UVmap (U project from view, bounds) for the face
then I created a simple material, shadeless.
added a texture (the mummy .png) and used the UV mapping method, specifying the UVmap created at point 4. This makes the .png fit perfectly the object face.
lastly I animated a full turnaround with using keyframes, then gave linear extrapolation.


If you wish to learn, try asking here about each of the above points that you don't understand (maybe try before because it could be really simple) just by referring to manual or other examples